Outside In

The computer animation Outside In
explains the amazing discovery,
made by Steve Smale in 1957,
that a sphere can be turned inside out by means of
smooth motions and self-intersections.
Through a combination of dialogue and exposition accessible to anyone
who has some interest in mathematics,
Outside In builds up to the grand finale:
Bill Thurston's
``corrugations'' method of turning the sphere inside out.
Along the way, the narrators discuss the related case of closed curves
and why they generally cannot be turned inside out.
Everyday analogies such as train tracks, belts, smiles and frowns are
used throughout, all richly animated and complete with sound
effects.

Graphics from Outside In:

About the video Outside In:

Software for everting spheres:

You can download the C++ program,
written by Nathaniel Thurston and used by the developers of
Outside In. It will generate a representation of the everting
sphere at any stage in terms of polygons or bicubic patches.